Hopefully, GENERAL is the correct section to post my two questions re The Galilean equation x' = x -vt.I will confine any possible future xkcd comments re this topic, to this one thread here in the General section.

wiki wrote:The notation below describes the relationship under the Galilean transformation between the coordinates (x,y,z,t) and (x′,y′,z′,t′) of a single arbitrary event, as measured in two coordinate systems S and S', in uniform relative motion (velocity v) in their common x and x’ directions, with their spatial origins coinciding at time t=t'=0:

x'=x-vt y'=y z'=z t'=t

1 x' as measured from (0,0,0) of S or S' or manifold M ?2 x as measured from (0,0,0) of S or S' or manifold M ?

"While statistics and measurements can be misleading, mathematics itself, is not subjective.""Be careful of what you believe, you are likely to make it the truth."steve

Steve, if you truly want to understand the Galilean transform, I'm willing to work you through it with you. But, it's not going to be quick and it's going to take quite a bit of work on your behalf. If you're willing to do that work, here's what'll happen.

I'll make a series of posts, each of which will define a few terms, work a few examples, and ask a few questions. You'll need to read each post carefully. Each time you find a term you don't understand or a definition that's unclear, you need to stop and ask for clarification. Do not continue until we're clear. You need to work each example yourself and ensure you get the same answer as I give. You need to solve each problem, using the definitions and notations I've introduced, and post your answers. Once I check your answers, I'll make my next post, and we'll repeat the process with that. I'm not quite sure how many steps this will take, but I'm guessing it'll be 5-10. Only at the end will we get to the Galilean transformation, so you'll need patience and endurance.

This is asking a lot, I know, so let me point out what I'm not asking: I'm not asking you to believe the definitions I give are "right" or the best to represent the world. I'm just asking you to use them as we go through and derive the Galilean transform, since they are the ones we need. Once we've done that, you're free to decide that these definitions are stupid and you can present your new system which is obviously better. But you won't be able to say the Galilean transformation is wrong.

Are you up for it? If so, respond here and I'll write up the first post.

Schrollini wrote:Steve, if you truly want to understand the Galilean transform, I'm willing to work you through it with you. But, it's not going to be quick and it's going to take quite a bit of work on your behalf. If you're willing to do that work, here's what'll happen.

I'll make a series of posts, each of which will define a few terms, work a few examples, and ask a few questions. You'll need to read each post carefully. Each time you find a term you don't understand or a definition that's unclear, you need to stop and ask for clarification. Do not continue until we're clear. You need to work each example yourself and ensure you get the same answer as I give. You need to solve each problem, using the definitions and notations I've introduced, and post your answers. Once I check your answers, I'll make my next post, and we'll repeat the process with that. I'm not quite sure how many steps this will take, but I'm guessing it'll be 5-10. Only at the end will we get to the Galilean transformation, so you'll need patience and endurance.

This is asking a lot, I know, so let me point out what I'm not asking: I'm not asking you to believe the definitions I give are "right" or the best to represent the world. I'm just asking you to use them as we go through and derive the Galilean transform, since they are the ones we need. Once we've done that, you're free to decide that these definitions are stupid and you can present your new system which is obviously better. But you won't be able to say the Galilean transformation is wrong.

Are you up for it? If so, respond here and I'll write up the first post.

Yes. Let's give it a try. Certainly, we need to agree upon what being a Cartesian x co-ordinate means mathematically, as part of any initial set of terms and definitions.

"While statistics and measurements can be misleading, mathematics itself, is not subjective.""Be careful of what you believe, you are likely to make it the truth."steve

steve waterman wrote:Yes. Let's give it a try. Certainly, we need to agree upon what being a Cartesian x co-ordinate means mathematically, as part of any initial set of terms and definitions.

Just to be clear, there's no issue of agreement at this point. We're going to work with the conventional definitions and show how that leads to the Galilean transform. At the end, we can debate whether what we did was the best thing to do or not, but not until then.

Let's get started.

Definition 1: A manifold is something that everywhere looks locally like Euclidean space. Wikipedia has a more detailed definition, but this should be good enough for our purposes.

For definiteness, let's consider only 2-dimesional manifolds, things that look locally like the Euclidean plane. The Euclidean plane is obviously a manifold. A sphere and a torus are also two dimensional manifolds, since they look at every point like a point in the plane. And since I just used the word, I ought to introduce

Definition 2: A point is an element in a manifold. If I want to say that the point P is an element of manifold M, I could write P ∊ M. (But I'll try to avoid doing that and just put things in English.)

If I have two points, P and Q, in a manifold M, the only thing I can say is whether P = Q or not. If they aren't the same point, I can't say anything about the distance or direction between them. The manifold doesn't contain such information. To get it, we need to introduce...

Definition 3: A coordinate system is a mapping between ℝn and a manifold M. That is, a function that takes n numbers (the coordinates) and returns a point P in the manifold M. The mapping is one-to-one, meaning that each set of coordinates maps to a single point in the manifold, and each point in the manifold has only one set of coordinates mapped to it. Not all coordinates need be mapped to the manifold, and not all points in the manifold need to have coordinates.

Since we're dealing with 2-dimensional manifolds, the coordinate system will take a pair of numbers to a point in the manifold. If I call the coordinate system f and the coordinates x and y, I can write f(x,y) = P ∊ M. The mapping being one-to-one means that f(x,y) = f(a,b) if and only if x = a and y = b.

Because the function f is one-to-one, I can invert it to find a function f-1 that maps M to ℝ2. That is, f-1(P) = (x,y) if and only if f(x,y) = P.

Excercise 1: Find f-1(f(x,y)).Excercise 2: Find f(f-1(P)).

This is all dry and boring, so let's have an example to finish this off. Let M be a manifold and f a coordinate system. By marking all the points f(x,0), we can put an x-axis on the manifold. Similarly, we can mark all the points f(0,y) to draw a y-axis. If we do this, we get something like this:

I am purposefully using unusual notation here to emphasize that the coordinates do not live in the manifold -- the images of the coordinates live in the manifold. This is why I've labeled everything in the form f(x,y).

Example: f(2,1) = P. Therefore, we say that the coordinates of P, in the coordinate system f, are (2,1).

Exercise 3: Draw the point f(-1, 2) = Q on the manifold above.Exercise 4: Find the coordinates of the point R marked above in the coordinate system f.

I think this is enough for one night. What I need you to do is read this through carefully. If you come to something unclear, stop and ask for clarification -- don't just make up a definition and continue on. Once everything's clear, go through the examples and see that you understand how I got these results. If you don't understand, ask for help. Then do the exercises and post your answers here. Once we're all on the same page, we'll go on to the next step: coordinate transformations. (That should be a bit more interesting that this.)

A note to others: Please don't answer the exercises I've posted for Steve. He needs to work through them for himself to be sure that he really understands the definitions I'm using. You're welcome to offer encouragement, hints, and wry commentary, but please let him do the work.

This is a nice start — props to you, Schrollini, for offering & taking the time to do this, and to you, Steve (I seem to recall you don't like being called 'Waterman'; please correct me if I'm mistaken) for agreeing to do it despite how confident with the material I know you feel you are.

cemper93 wrote:Dude, I just presented an elaborate multiple fraction in Comic Sans. Who are you to question me?

Let's wait and see what steve's response is before giving him props. If this goes the way I expect it to, you'll wish that I was right, and that this was a thread where we posted our favorite bits of nonsense poety. (Prove me wrong, steve. Please.)

re - terms; attempting to agree upon definitions for manifold M and a point

Schrollini wrote:Definition 2: A point is an element in a manifold. If I want to say that the point P is an element of manifold M, I could write P ∊ M. (But I'll try to avoid doing that and just put things in English.)

If I have two points, P and Q, in a manifold M, the only thing I can say is whether P = Q or not.If they aren't the same point,

I am confused by your statement, you just said that we have two points P and Q, yet now you say they can be the same point?...do you in fact mean, that point P and point Q can share the same location in manifold M?

Schrollini wrote:I can't say anything about the distance or direction between them. The manifold doesn't contain such information.

So... by itself, manifold M has no assigned location for f(0,0,0)?So, by itself, manifold M has no inherently assigned co-ordinates, and in our case, only has a point P and a point M that share the same location wrt manifold M?Question please - given system f(x,y,z), what can be said about the manifold M wrt f?

Also please, points are only ever IN manifold M?

"While statistics and measurements can be misleading, mathematics itself, is not subjective.""Be careful of what you believe, you are likely to make it the truth."steve

steve waterman wrote:I am confused by your statement, you just said that we have two points P and Q, yet now you say they can be the same point?...do you in fact mean, that point P and point Q can share the same location in manifold M?

No he doesn't; he never said they were two different points, just two points. He's establishing that you can only tell whether Point P and Point Q are the same point (a rose by any other name etc.), or different points. As for the rest of it, I'll leave it to Schrollini since it's his lessons and I've no idea how fast he wants to take it.

Edit: I just realised, I can't remember your pronouns, Schrollini; am I using the correct ones?

cemper93 wrote:Dude, I just presented an elaborate multiple fraction in Comic Sans. Who are you to question me?

steve - Schrollini is trying to teach you. Please at least be polite and follow along and do the exercises, rather than jumping around.

Exercise 1: Find f-1(f(x,y)).Exercise 2: Find f(f-1(P)).

Example: f(2,1) = P. Therefore, we say that the coordinates of P, in the coordinate system f, are (2,1).

Exercise 3: Draw the point f(-1, 2) = Q on the manifold above.Exercise 4: Find the coordinates of the point R marked above in the coordinate system f.

As he suggested, we will get to the exercises only AFTER we agree upon what our terms are defined as.If my one personal question cannot be answered now, then it too will have to wait. My question does not contain any of the terms under current scrutiny, and thus I thought someone might have the ability to answer it.

steve waterman wrote:As he suggested, we will get to the exercises only AFTER we agree upon what our terms are defined as.If my one personal question cannot be answered now, then it too will have to wait. My question does not contain any of the terms under current scrutiny, and thus I thought someone might have the ability to answer it.

steve waterman wrote:As he suggested, we will get to the exercises only AFTER we agree upon what our terms are defined as.If my one personal question cannot be answered now, then it too will have to wait. My question does not contain any of the terms under current scrutiny, and thus I thought someone might have the ability to answer it.

As you yourself state, none of the terms you are using has yet been defined - no one should answer them until we all agree on the definitions.

Schrollini suggested this notation, which I have employed above - some system letter ( abscissa, ordinate, applicate ) as would be S(x,y,z) or f(a,b,c).What term(s) are you unclear as to their exact mathematical meaning?

"While statistics and measurements can be misleading, mathematics itself, is not subjective.""Be careful of what you believe, you are likely to make it the truth."steve

Steve, you seem to have missed the following parts of schrollini's original post:

You'll need to read each post carefully. Each time you find a term you don't understand or a definition that's unclear, you need to stop and ask for clarification. Do not continue until we're clear. You need to work each example yourself and ensure you get the same answer as I give. You need to solve each problem, using the definitions and notations I've introduced, and post your answers.

...I'm not asking you to believe the definitions I give are "right" or the best to represent the world. I'm just asking you to use them as we go through...

What he proposed was not a dialog where you attempted to agree upon definitions. By all means, ask for clarification in any instances where you don't understand what his terms mean. But sticking to addressing the specifics of his posts would be very helpful. None of what you're asking was part of his first "teaching" post.

JudeMorrigan wrote:Steve, you seem to have missed the following parts of schrollini's original post:

You'll need to read each post carefully. Each time you find a term you don't understand or a definition that's unclear, you need to stop and ask for clarification. Do not continue until we're clear. You need to work each example yourself and ensure you get the same answer as I give. You need to solve each problem, using the definitions and notations I've introduced, and post your answers.

...I'm not asking you to believe the definitions I give are "right" or the best to represent the world. I'm just asking you to use them as we go through...

What he proposed was not a dialog where you attempted to agree upon definitions. By all means, ask for clarification in any instances where you don't understand what his terms mean. But sticking to addressing the specifics of his posts would be very helpful. None of what you're asking was part of his first "teaching" post.

That is why I asked him questions, yet unanswered. For example, does manifold M have a fixed (0,0,0).I need to know first, BEFORE we discuss any ramifications, what we mean by our starting terms. I did not just say the manifold M is questioned, I asked if it has one and only one possible location for it's (0,0,0).

Can I not ask one personally based question while we figure out what is meant by manifold M? That seems pretty reasonable to me. It does not even need to be Schrollini to give your own opinion.

x' = x-vt; x transforms to x or to x' ?

"While statistics and measurements can be misleading, mathematics itself, is not subjective.""Be careful of what you believe, you are likely to make it the truth."steve

Oh, I think your first post of questions was fine, and I imagine schrollini will be happy to answer once he gets a chance. I just think your "personally based question" is mis-timed and that we'll all be better off sticking with the proposed exercise.

Woah, woah, woah, everybody! Settle down and take a chill pill. Steve did just as I asked in not going on to the exercises while there were still open questions. But I hope he understands that I'm not going to answer his question yet. We have miles to go before we get there!

yurell wrote:

steve waterman wrote:I am confused by your statement, you just said that we have two points P and Q, yet now you say they can be the same point?...do you in fact mean, that point P and point Q can share the same location in manifold M?

No he doesn't; he never said they were two different points, just two points. He's establishing that you can only tell whether Point P and Point Q are the same point (a rose by any other name etc.), or different points.

yurell's answer is is correct. Allow me to say the same thing in other words: If P and Q each label a point in the manifold, I can tell whether they're labeling the same point or different points. It's like saying "the number one more than seven" and "the number one less than nine". Both of these refer to 8, they're just different names for it. The fact that I have two names for 8 doesn't mean I have two different numbers 8; I just have two names for it.

This is a good question, and I was sloppy on this point. Thanks for getting me to clarify.

steve waterman wrote:So... by itself, manifold M has no assigned location for f(0,0,0)?

Correct. The manifold M doesn't need a coordinate system. It can exist perfectly happily just by itself.

Also, note that a coordinate system f may not necessarily map (0,0) to a point in the manifold. Recall that in the definition, I specified that f maps a subset of ℝ2 to a subset of M. That subset of ℝ2 doesn't not have to include (0,0).

steve waterman wrote:So, by itself, manifold M has no inherently assigned co-ordinates, and in our case, only has a point P and a point M [sic - perhaps you mean Q?] that share the same location wrt manifold M?

Yes to the first part. Coordinates are external to the manifold. The manifold contains an uncountable infinity of points, so I can't label them all. P and Q are just two that I've happened to label. They may or may not be the same point, but that's all I can say about them. I can't say that P is west of Q or that Q is 4 inches from P.

steve waterman wrote:Question please - given system f(x,y,z), what can be said about the manifold M wrt f?

With the coordinate system, I can start saying things about the distance and direction between points. A manifold with a coordinate system is an example of a metric space, which show up all over mathematics and physics. But I don't really want to get into the details of that now. The important thing for the time being is that a coordinate system maps coordinates to points in a manifold.

steve waterman wrote:Also please, points are only ever IN manifold M?

Points are elements of a manifold. There are many manifolds, but for this exercise, we'll only consider the Cartesian plane, if that helps.

I hope this clarifies things. If so, go onto the example and exercises. But if not, don't hesitate to ask more. Just know that I may take hours to respond, so please don't get antsy.

yurell wrote:Edit: I just realised, I can't remember your pronouns, Schrollini; am I using the correct ones?

Schrollini wrote:Woah, woah, woah, everybody! Settle down and take a chill pill. Steve did just as I asked in not going on to the exercises while there were still open questions. But I hope he understands that I'm not going to answer his question yet. We have miles to go before we get there!

That's all I was intending to say, that the "personal question" was premature and that I thought it would be better if we did things in order and didn't get side-tracked.

Schrollini wrote:Also please, points are only ever IN manifold M?Points are elements of a manifold. There are many manifolds, but for this exercise, we'll only consider the Cartesian plane, if that helps.

It does help.

So, your manifold M has? 1 NO co-ordination at all2 Moves as the moved system does3 Contains points as elements 4 So these points of manifold M are not deemed as "co-ordinates", rather as "elements" wrt manifold M.5 SO all manifold M elements are points and none of those points have associated (x,y,z) evaluations.

"While statistics and measurements can be misleading, mathematics itself, is not subjective.""Be careful of what you believe, you are likely to make it the truth."steve

Coordinates are not inherent to the manifold, if that's what you're asking. They are only associated with a coordinate system.

steve waterman wrote:2 Moves as the moved system does

Nothing's moving. There are no dynamics. The manifold simply is. The coordinate system simply is.

I think you're trying to see how this will connect to the Galilean transformation. All I can say is, patience. We'll get to it, but it's going to take a while.

steve waterman wrote:3 Contains points as elements 4 So these points of manifold M are not deemed as "co-ordinates", rather as "elements" wrt manifold M.5 SO all manifold M elements are points and none of those points have associated (x,y,z) evaluations.

Yes, yes, and yes. The distinction between the manifold and the coordinate system is absolutely critical, so I'm glad you're taking care in understanding the difference.

I'm not sure precisely what you mean by this, but I have a feeling that it's not important at this exact instant, and that Schrollini will nail this down to your satisfaction at a later time.

steve waterman wrote:3 Contains points as elements 4 So these points of manifold M are not deemed as "co-ordinates", rather as "elements" wrt manifold M.5 SO all manifold M elements are points and none of those points have associated (x,y,z) evaluations.

Correct. The manifold M is a set of points. We can use any coordinate system we like to label the points and make sense of them, but the points themselves do not require a coordinate system to exist. Points in a manifold are just things in a set.

I like what's happening here. Thanks to Schrollini for being willing to take this on, and thanks to Steve for being willing to go through this process. I feel that I could also learn some things by going through a rigorous definition of all the fiddly details my math teachers glossed over.

Schrollini wrote:Nothing's moving. There are no dynamics. The manifold simply is. The coordinate system simply is.

Thanks for clearing this up, as I though the manifold was movable.

Okay, Manifold M is stationary wrt t.

Could we not use the term..."manifold points" to mean points that are the elements of manifold M, andthus, have no need for the superficial term, "elements" conceptually?

I have for a long long time referred to "co-ordinate points" and now see why that would be most confusing here at xkcd. I will totally drop this term and use "co-ordinates" to mean wrt to some specific (0,0,0).

I think I have enough for manifold M and point ( a working understanding ), I will check out your thoughts on co-ordinate systems next. Break time, for a few hours or so. Nice exchange.

"While statistics and measurements can be misleading, mathematics itself, is not subjective.""Be careful of what you believe, you are likely to make it the truth."steve

Schrollini wrote:Since we're dealing with 2-dimensional manifolds, the coordinate system will take a pair of numbers to a point in the manifold. If I call the coordinate system f and the coordinates x and y, I can write f(x,y) = P ∊ M.

Little lost with the notation on this one...Would it be better said that say, f(2,0) = P ∊ M, since P is at a unique location wrt M,otherwise P would be every (x,y) in M using f(x,y) = P ∊ M?

"While statistics and measurements can be misleading, mathematics itself, is not subjective.""Be careful of what you believe, you are likely to make it the truth."steve

SecondTalon wrote:Can we, as a group, do our best to keep this the Schrollini and Steve show?

..in other words, if you ain't Schrollini or Steve.. why you replyin' like you are?

I was mere moments away from posting this, but ST is faster than a speeding e-mail. Schrollini seems more than amply qualified to do this solo, and others interjecting will lead to side comments and diversions. Best to let Schrollini work his magic until he cannot or chooses to stop. Of course, by posting this, I'm not following my own advice. So I'll try to stop now.

Advice to Steve: Please try your best to answer Schrollini's questions only, and not necessarily questions about what you think the implications are. Schrollini seems to have impressive patience; I'm sure you'll get to the good stuff when the solid foundation is finished.

Don't think about time yet. There is no time. I will eventually introduce it, but the way I do will probably not be the way you're expecting. So please don't try to fit time in now -- you'll only confuse yourself later on.

steve waterman wrote:Could we not use the term..."manifold points" to mean points that are the elements of manifold M, andthus, have no need for the superficial term, "elements" conceptually?

Element is a standard set theoretic term. A collection of elements makes up a set, and the components of a set are elements. I used this term because a manifold is a set, and the points of the manifold are the elements of the set. Now that we have an understanding of manifolds, I'll try not to use these terms again. We'll just use "points" and "manifolds".

steve waterman wrote:I have for a long long time referred to "co-ordinate points" and now see why that would be most confusing here at xkcd. I will totally drop this term and use "co-ordinates" to mean wrt to some specific (0,0,0).

Exactly!

steve waterman wrote:Would it be better said that say, f(2,0) = P ∊ M, since P is at a unique location wrt M,

This is certainly true.

steve waterman wrote:otherwise P would be every (x,y) in M using f(x,y) = P ∊ M?

What I mean to say is that, given a specific x and y, f(x,y) is a point in the manifold. If I want to give it a label, I could call it P. If you choose two other numbers, a and b, f(a,b) will be another point in the manifold, which I might call Q.

I'm a bit inconsistent later, when I use the expression f(x,0) to indicate the points for all values of x. What I should write there is {f(x,0) | ∀ x ∊ ℝ}, which is math-speak for "the set of points f(x,0) for all real numbers x". But both of these seemed too long to write, so I abbreviated.

SexyTalon wrote:Can we, as a group, do our best to keep this the Schrollini and Steve show?

..in other words, if you ain't Schrollini or Steve.. why you replyin' like you are?

Yes, I would appreciate that a lot. I do have this one exception that I would like to hear a few voices upon, please

x' = x-vt; x transforms to x or to x' ?

All I wish to hear, is your answer, and not any logic to support it.

You're jumping ahead. Be patient. The original problem with the Pressures thread was competition between standard terminology and concepts and Stevean math, and continuing that competition as is serves no one. Let Schrollini help you get the right terminology and concepts, and the answer to your question will come in time. But if you insist on jumping to an ill-defined question based on your incorrect understanding of the subject, you'll get the same confusion that torpedoed the Pressures thread.

Insanity is doing the same thing over and over again, expecting a different result. Are you insane, Steve?

coordinate S(2,3,4) is/represents a mathematical point. Do you agree?Some points are coordinates, therefore saying that all points are in the manifold is confusing/inaccurate.

I suggest we employ these terms? to differentiate two different types of points we have the set of "manifold points"we have the infinite set of "coordinate points"

Both ARE point sets. I am quite familiar with point sets as they are basis for generating waterman clusters and resultant polyhedra.

So, I am not willing to accept the term "point" without its qualifier...(which type of point), Well, that is, only if you agreed that S(2,3,4) represents a mathematical point.

ADDED -

A Cartesian coordinate system is a mathematical tool used in plotting points and graph lines. It is a system in which the location of a point is given by coordinates that represent its distances from perpendicular lines that intersect at a point called the origin.

steve waterman wrote:All "points" are in the manifold; "manifold points".

All real numbers are in the number line, but we don't call them "number line numbers", because that would be stupid. Similarly, all points live in a manifold. We don't need to call them "manifold points" because there will be no other types of points in this discussion.

If you find this confusing, you may mentally replace point with "element of the manifold".

No no no no, a thousand times no! Points live in the manifold. Coordinates live in ℝ2. They are related only in that a coordinate system maps coordinates to points. Sometimes we may be sloppy and say that a point P has coordinates (x,y). But what we're really saying is that f(x,y) = P; that is, the coordinate system maps (x,y) to the point P.

It's only natural to try to fit these concepts into your existing mental framework, but I'm asking you not to do this. Just work with these few definitions and forget that they are connected to anything larger. At the end, we can go back and see what can and can't be translated into your language. But for now, we have a manifold with points, coordinates in ℝ2, and a coordinate system that maps coordinates to points. Nothing else.

Quote mining is going to get us nowherea and wasn't what the exercise was about. But I can't resist the temptation (although I suspect it's going to annoy some of the other innocent bystanders) to note that you're misreading some of your quotations. The University of Utah quote, for example, state that coordinates are "the location of a point". That is, they are not the point itself. The quote is the functional equivalent of Schrollini's "Sometimes we may be sloppy and say that a point P has coordinates (x,y)."

Again, I'm going to encourage you to stick with addressing Schrollini's posts and the content therein directly. (ETA: Note that that does, in fact, mean that I'm recommending you not respond to me. Instead, just focus on "do I understand what Schrollini's writing? Does it make internal sense?" Note that "does it agree with my own preconceptions? Does it agree with what I think I've read elsewhere?" are questions that I'm encouraging you to table. Not set aside entirely. Just table for now.)

In my freshman math course, Prof. Brin (father of Sergey (how's that for namedropping!)) would occasionally poll us as to what we thought the answer to a problem was. He would tally the votes, reveal that most of us were wrong, and quip, "It's a good thing math isn't a democracy."

My point is, I don't care how many websites you quote. We're working with my definitions (which are more-or-less conventional) only.

But let me offer three reasons why you may find apparently contradictory notions of points:

The first, as JudeMorrigan points out, is that the author may be sloppy. Right now, I'm being (or at least trying to be) very careful with my notation so that we can understand the differences between points and coordinates. I actually plan on introducing sloppier notation shortly, but first we have to make sure we're all on the same page. Seeking out writers that are not being as careful as I isn't helping us.

The second is that ℝ2 is a manifold itself, so it's perfect reasonable to talk about the elements of ℝ2 being points. But this has the potential to be confusing, so I will never make use of ℝ2 being a manifold, and I will only talk about points that are in a manifold M which is the range of a coordinate system. So try to forget this paragraph.

The third is that the author is wrong, or to put it more gently, is using a different set of definitions that I am. That's okay, since there's no "right" set. I'm using a set of definitions I know will lead to the Galilean transformation, but I'm not going to check that every other set of definitions you may find will also work.

In short, please ask questions if a definition is unclear. But if the definition is just unpleasant to you, you have to grit your teeth and power through. As I said, at the end you'll be free to decide that my definitions are stupid, useless, ugly, whatever. But until then, you have to use them.

Schrollini wrote:In short, please ask questions if a definition is unclear. But if the definition is just unpleasant to you, you have to grit your teeth and power through. As I said, at the end you'll be free to decide that my definitions are stupid, useless, ugly, whatever. But until then, you have to use them.

So, you insist that coordinates are not associated with points...fine, I accept that that is what you believe that coordinates are not points.

Now tell me about mapping, please, in your own words. If you could also comment too please.

Imagine having no points and t = 0....GIVEN/LET only one system, S(x,y,z)....for you, S requires mapping for S to contain all the coordinates, apparently.

"While statistics and measurements can be misleading, mathematics itself, is not subjective.""Be careful of what you believe, you are likely to make it the truth."steve

A mapping, or a function*, from one set to another associates elements of the first set with elements of the second. If f maps set A to set B and a ∊ A, then f(a) ∊ B. A mapping is one-to-one if each element of A is associated with a single element of B, and each element of B is associated with (at most) a single element of A.

In our case, the coordinate system is a mapping from the set ℝ2** (whose elements are the coordinates) to the manifold M (whose elements are the points). Thus, if f is a coordinate system and (x,y) ∊ ℝ2, then f(x,y) ∊ M.

* Some people make a distinction that a function is a specific type of mapping. But since any one-to-one mapping is a function, I'll ignore this distinction.

** Properly speaking, the coordinate system is a mapping from a subset of ℝ2 to M, since not all coordinates need to represent a point in the manifold. But we can generally ignore this complication.

steve waterman wrote:Imagine having no points and t = 0....

I'm not addressing this, and I think you know why.

Edit to add:Sorry, I think I misunderstood your last point. But please, don't mention time anymore. There is no time (yet).

Anyway, if there are no points, there is no manifold. If there's no manifold, there's no mapping to that manifold. And therefore there's no coordinate system. So your question really doesn't make sense.

Schrollini, when you are listing functions and letters by themselves (or even together), it may be a good idea to italicise them; it's hard to distinguish f from a typo (or just skip over it entirely with your eye), but f is somewhat more obvious. It also creates a solid break between mathesese and general text that I find makes it easier to read.

cemper93 wrote:Dude, I just presented an elaborate multiple fraction in Comic Sans. Who are you to question me?